Problem: Simplify; express your answer in exponential form. Assume $q\neq 0, x\neq 0$. $\dfrac{{(q^{-4}x^{-3})^{2}}}{{(q^{-1}x^{5})^{2}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-4}x^{-3})^{2} = (q^{-4})^{2}(x^{-3})^{2}}$ On the left, we have ${q^{-4}}$ to the exponent ${2}$ . Now ${-4 \times 2 = -8}$ , so ${(q^{-4})^{2} = q^{-8}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-4}x^{-3})^{2}}}{{(q^{-1}x^{5})^{2}}} = \dfrac{{q^{-8}x^{-6}}}{{q^{-2}x^{10}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-8}x^{-6}}}{{q^{-2}x^{10}}} = \dfrac{{q^{-8}}}{{q^{-2}}} \cdot \dfrac{{x^{-6}}}{{x^{10}}} = q^{{-8} - {(-2)}} \cdot x^{{-6} - {10}} = q^{-6}x^{-16}$